Bob Coecke- Quantum Picturalism

8/3/2019 Bob Coecke- Quantum Picturalism

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Quantum Picturalism

(or, The Logic of Quantum Mechanics take 2)

PSA Montreal November 2010

=

f

f=

f f

f

ALICE

BOB

=

ALICE

BOB

f

=

not

like

BobAlice

does

Alice not like

not

Bob

Bob CoeckeOxford University Computing Laboratory


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The Hilbert space quantum formalism


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[von Neumann 1932] Formalized quantum mechanics

in Mathematische Grundlagen der Quantenmechanik


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[von Neumann to Birkhoff 1935] I would like to

make a confession which may seem immoral: I do not

believe absolutely in Hilbert space no more. (sic)


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[Birkhoff and von Neumann 1936] The logic of Quan-

tum Mechanics, Annals of Mathematics.


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[1936 2000] many followed them, ... and FAILED.


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[1936 2000] many followed them, ... and FAILED.


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Hilber space stuff: continuum, field structure of com-

plex numbers, vector space over it, inner-product, etc.


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WHY?


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WHY?

von Neumann: only used it since it was available.


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WHY?


Model theory: one can do almost anything with it.


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WHY?


Model theory: one can do almost anything with it.

Schrodinger (1935): the stuff which is the true soul of

quantum theory is how quantum systems compose.


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tensor product structurethe other stuff

= ?


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= ?

Conceptually: not about properties of the individual,

but about relationships among the individuals


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= ?



Mathematically: axiomatize an abstract tensor prod-

uct without reference to underlying spaces


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= ?





1. Game plan: Which assumptions (i.e. which struc-

ture) on is needed to deduce physical phenomena?


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= ?





1. Game plan: Which assumptions (i.e. which struc-

ture) on is needed to deduce physical phenomena?

2. Additional question: Does such an interaction struc-

ture appear elsewhere in our classical reality?


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Outcome 1a: Sheer ratio of results to assumptions

Hans Halvorson (2010) Editorial to: Deep Beauty: Understanding the Quan-

tum World through Mathematical Innovation, Cambridge University Press.


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confirms that we are probing something very essential.

Hans Halvorson (2010) Editorial to: Deep Beauty: Understanding the Quan-

tum World through Mathematical Innovation, Cambridge University Press.


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Outcome 1b: Exposing this structure has already helped

to solve open problems elsewhere. (e.g. 2 ICALP10)

EG: Ross Duncan & Simon Perdrix (2010)Rewriting measurement-based quan-

tum computations with generalised flow. ICALP10.


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Outcome 1c: Simple intuitive (but rigorous) dia-grammatic language, meanwhile adopted by others:

... we join the quantum picturalism revolution [1]

Lucien Hardy (2010) A formalism-local framework for general

probabilistic theories including quantum theory. arXiv:1005.5164

[1] Coecke (2010) Quantum picturalism. Contemporary Physics 51, 5983.

arXiv:0908.1787 (survey)


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Outcome 1c: Simple intuitive (but rigorous) dia-grammatic language, meanwhile adopted by others:

... we join the quantum picturalism revolution [1]

Lucien Hardy (2010) A formalism-local framework for general

probabilistic theories including quantum theory. arXiv:1005.5164

[1] Coecke (2010) Quantum picturalism. Contemporary Physics 51, 5983.

arXiv:0908.1787 (survey)


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Outcome 2a:Behaviors of matter:

=

f

f =

f f

f

ALICE

BOB

=

ALICE

BOB

f


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=

f

f =

f f

f

ALICE

BOB

=

ALICE

BOB

f

Meaning in language:

=

not

like

BobAlice

does

Alice not like

not

Bob

meaning vectors of words

pregroup grammar


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=

f

f =

f f

f

ALICE

BOB

=

ALICE

BOB

f

Meaning in language:

=

not

like

BobAlice

does

Alice not like

not

Bob


pregroup grammar

Knowledge updating :

conditionalindependence

=P(C|AB)

A A

=

A

=

A

B

A

B

=

B

(BA) 1-A

C 1- C 1-

C

P(AB|C) P(A|C) P(B|C) P(C|A) P(C|B)

P(C|B) P(C|A)


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Outcome 2b: The structure is a true (quantum) logic:

- I can give you a demo backstage -


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A MINIMAL LANGUAGE

FOR QUANTUM REASONING

Abramsky & Coecke (2004) A categorical semantics for quantum protocols.arXiv:quant-ph/0402130

Coecke (2005) Kindergarten quantum mechanics.

arXiv:quant-ph/0510032


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(physical) data in the language

Systems:

A B C

Processes:

Af-A A

g-B B

h-C

Compound systems:

A B I A Cfg

-B D

Temporal composition:

Ahg

-C := Ag-B

h-C A

1A-A


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graphical notation

g f

g

f

f g f fg

Roger Penrose (1971) Applications of negative dimensional tensors.

In: Combinatorial Mathematics and its Applications. Academic Press.

Andre Joyal & Ross Street (1991) The geometry of tensor calculus I.

Advances in Mathematics 88, 55112.


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merely a new notation?

(g f) (k h) = (g k) (f h)

=

f h

g k

f h

g k


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merely a new notation?

(g f) (k h) = (g k) (f h)

=

f h

g k

f h

g k

peel potato and then fry it,

while,

clean carrot and then boil it=

peel potato while clean carrot,

and then,

fry potato while boil carrot


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graphical notation

: I A : A I : I I

A

A


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graphical notation

: I A : A I : I I

A

A


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graphical notation

: I A : A I : I I

A

A


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graphical notation

: I A : A I : I I

A

A


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graphical notation

: I A : A I : I I

A

A


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graphical notation

: I A : A I : I I

A

A


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graphical notation

: I A : A I : I I

A

A


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graphical notation

: I A : A I : I I

A

A


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graphical notation

: I A : A I : I I

A

A


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graphical notation

: I A : A I : I I

A

A


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adjoint

f : A B

f

A

B


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adjoint

f : B A

f

B

A


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asserting (pure) entanglement

quantum

classical=

=

=


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quantum-like

A A


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quantum-like

A A

=

A

A

A

A


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quantum-like

A A

=

A

A

A

A


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quantum-like

A A

=

A

A

A

A


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quantum-like

A A

=

A

A

A

A

lik


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quantum-like

A A

=

A

A

A

A

lik


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quantum-like

ff=

lidi


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sliding

=f f

=f

f

lidi


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sliding

=f f

=f

f

In QM: cups = Bell-states, caps =Bell-effects, -rotations = transpose

classical data flow?


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f=

f

ff



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f

=

f



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f

=

f



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f

ALICE

BOB

=

ALICE

BOB

f

quantum teleportation

Applying decorated normalization 3


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Applying decorated normalization 3

=

f

f

f f

Entanglement swapping



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f

=

f

g

g

gate teleportation computation

dagger compact categories


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Thm. [Kelly-Laplaza 80; Selinger 05] An equa-

tional statement between expressions in dagger com-

pact categorical language holds if and only if it is

derivable in the graphical notation via homotopy.



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gg p g

Thm. [Kelly-Laplaza 80; Selinger 05] An equa-

tional statement between expressions in dagger com-

pact categorical language holds if and only if it is

derivable in the graphical notation via homotopy.

Thm. [Selinger 08] An equational statement between

expressions in dagger compact categorical language

holds if and only if it is derivable in the category of

finite dimensional Hilbert spaces, linear maps, tensorproduct, andadjoints.



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gg p g

In words: Any equation involving:

states, operations, effects

unitarity, adjoints (e.g. self-adjoint), projections

Bell-states/effects, transpose, conjugation

inner-product, trace, Hilbert-Schmidt norm

positivity, completely positive maps, ...

holds in quantum theory if and only ifit can be derived

in the graphical language via homotopy.


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A SLIGHTLY DIFFERENT LANGUAGE

FOR NATURAL LANGUAGE MEANING

Coecke, Sadrzadeh & Clark (2010) Mathematical Foundations for a Compo-

sitional Distributional Model of Meaning.

arXiv:1003.4394

the from-words-to-a-sentence process


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Consider meanings of words (e.g. vectors as in Google):

word 1 word 2 word n

...

?

How do we/machines compute meaning of sentences?



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Consider meanings of words (e.g. vectors as in Google):

word 1 word 2 word n

...

grammar

How do we/machines compute meaning of sentences?



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Information flow within a verb:

verb

object subject



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Information flow within a verb:

verb

object subject

Again we have:

=

going non-symmetric


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I

l

A Al

Al

Al

I I

r

Ar

A A Ar

r

I

A Al A A

A Al

r

A Ar

=

A

A

A

A

=A

A A

A

r

r

=

A

A

A

A

=A

A A

All

ll

r

r

Alice

does

not

like

Bob


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Alice

does

not

like

Bob


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Alice not like Bob


not

grammar

does

Alice

does

not

like

Bob


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Alice like Bob


grammar

not

Alice

does

not

like

Bob


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Alice like Bob


grammar

not

Alice

does

not

like

Bob


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Alice like Bob


grammar

not

= not

like

BobAlice

grammar


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Alice hates Bob


g

f

f

states

measurements

analogy: non-local info-flows English (& French):


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English (& French):

Hindi:

Persian:

Arabic (and Hebrew):

Mehrnoosh Sadrzadeh (2008) Pregroup analysis of Persian sentences.


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THE EXTENDED LANGUAGE:

COMPLEMENTARITY & CLASSICALITY

observables


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observables m


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spiders =

m

....

....

n

such that, for k > 0:

m+mk

........

....

....

....

n+nk

=

....

....

Coecke, Pavlovic & Vicary (2006, 2008) quant-ph/0608035, 0810.0812

observables


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Theorem 1. In any dagger symmetric monoidal cate-

gory families of spiders and dagger special commuta-

tive Frobenius algebra are in bijective correspondence.

observables m


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spiders =

m

....

....

n


m+mk

........

....

....

....

n+nk

=

....

....

observables


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Theorem 1. In any dagger symmetric monoidal cate-

gory families of spiders and dagger special commuta-

tive Frobenius algebra are in bijective correspondence.

Theorem 2. (Coecke-Pavlovic-Vicary) InFdHilb dag-

ger special commutative Frobenius algebra are exactlyorthonormal bases, namely those of copyable elts.

Coecke & Pavlovic (2007) Quantum measurement without sums. In: Mathe-

matics of Quantum Computing and Technology. quant-ph/0608035

Coecke, Pavlovic & Vicary (2008) A new description of orthogonal bases.

Mathematical Structures in Computer Science. 0810.0812

observables m


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spiders =

m

....

....

n


m+mk

........

....

....

....

n+nk

=

....

....

Coecke, Pavlovic & Vicary (2006, 2008) quant-ph/0608035, 0810.0812

complementarity


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complementarity


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Coecke & Duncan (2008)Interacting quantum observables. arXiv:0906.4725


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environment


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ground =

f=

g f

=

g

f g

Thm. mixed states, CP maps, class. probabilities.

Coecke & Perdrix (2010)Environment and class. chan. ... arXiv:1004.1598

environment


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ground =

f=

g f

=

g

f g

Thm. mixed states, CP maps, class. probs in Hilb.


environment


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ground =

f=

f

=f

Thm. mixed states, CP maps, class. probabilities.


Prop 1:


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=

Prop 2:

=

Destructive measurement:


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=

Non-destructive measurement:

= =

Indeed measurement:


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=

Indeed controlled unitary:

=

key distribution


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=

=

key distribution


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=

=

phase groups and universality for QC

Translation to circuits and determinism for so-called


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measurement based quantum computations:

H

H

H

, {3}

, {2}

, {2}

, {3}, {2}

-

H

H

H

, {3}

, {2}

,

{2}, {2} , {3}

-

H

H

H

, {3}

, {3}

, {2}

, {2}

, {2}

-

, {2}

, {2}

, {2}

-

, {2}, {2}

, {2} , {2}

-

Ross Duncan & Simon Perdrix (2010) Rewriting measurement-based quantum

computations with generalised flow. ICALP10.

phase groups and quantum non-locality

Toy qubits vs. true quantum theory in one language:


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Spekkens qubit QM

stabilizer qubit QM=

Z2 Z2Z4

=local

non-local

Bob Coecke, Bill Edwards & Rob Spekkens (2010) Phase groups and the ori-

gin of non-locality for qubits. arXiv:1003.5005

entanglement classification

Tripartite SLOCC-classes as comm. Frobenius algs:


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GHZ = |000 + |111W = |001 + |010 + |100

=special CFAs

anti-special CFAs

=

=

=

Coecke & Aleks Kissinger (2010) The compositional structure of multipartite

quantum entanglement. ICALP10. arXiv:1002.2540


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PUNCHLINE ON LOGIC

Taking (de)composition as primitive (vs. an individual

entities properties) we get very far with very little!


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105/107

An interaction logic is also present in natural lan-guage (= also the source of static orthodox logic).




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Overall stance: Quantum richness is not going to be

understood by weakening standard logical tools, but

by considering radically different (classical?) ones.




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Overall stance: Quantum richness is not going to be

understood by weakening standard logical tools, but

by considering radically different (classical?) ones.

Logic indeed (vs. quantum (non-)logic): Automa-

tion is demonstrated via the quantomatic software.

Dixon, Duncan & Kissinger. http://dream.inf.ed.ac.uk/projects/quantomatic/

Documents

Bob Coecke- Quantum Picturalism